Comet vs asteroid

Meteoroid - Small rocky, iron or icy debris flying in space. From microns to 1 meter.

Asteroid - A large meteoroid. From 1 meter to hundreds of kilometers.

Comet - A solid body made of ice, rock, dust and frozen gases. As they fracture and disintegrate, some Comets leave a trail of solid debris. Nucleus (solid part) from 16-60 km. Tail hundreds of millions of km.

So it seems to only difference between asteroids and comets by this definition is composition; more rock and less ice in an asteroid.

So my two questions are:

1. are there many (any) bodies with the composition of comets in the asteroid belt?

2. are there many bodies with the composition of asteroids in very elliptical, close perihelion orbits like comets?